g28 DR WALLACE ON A FUNCTIONAL EQUATION. 



10. We may consider this differential equation, and the functional equation 



/(*o)/(*.) = c {f{x^ + x) + fix, - x) } 

 as the representatives of each other, so that if x and y be co-ordinates of a curve, 

 the functional equation will express a property of that curve. Now, considering 

 ?/ as a function of x, the function and its variable may either increase together, 

 or else y may decrease while x increases : therefore, it may be, that the function 

 which satisfies the equation (A) will have different forms ; for if y decrease while x 



/J 'ij 



increases, the differential coeflBcient -r- will be negative ; if, however, y and x 



increase together, then it wiU be positive. 



11. Let us first consider the case in which y decreases while x increases. The 



differential equation to be resolved may then be expressed thus, (putting & for a 



constant), 



d^y 1 1 



dx^ y 



,3. 



^ m = 



__ydx . 



and, multiplying both sides by -j 



iX 



dy 

 dx 



\dx I c- 



and taking the integi-als. 



( 



dx) (? 



dv 



Now, -f- expresses the tangent of the angle which a line touching the 

 curve makes with the axes to which ^/ is a perpendicular ordinate : and since 



— manifestly cannot exceed h, therefore y must have a maximum value, which 



d y 

 must satisfy the equation -^ = 0, and putting a for that maximum value of y, 



we have 6--^=0: and ^--t-> therefore, \-j^)~ — ^i — ' 



dy 



and 



dx _ dy _ a 



from this, by integration, we get 



We may assume that, when ?/ is a maximum, and =a, then x—^\ therefore 



cosa = l, andsina = 0. 



